Warning

Scalars:

A quantity having only magnitude is called a scalar.Ex: Mass, length, temperature, area, volume etc. We identify real numbers also as scalars.

Directed line segment:

A line segment with an arrowhead showing direction is called a directed line segment. Its two endpoints are distinguished as initial (or tail) and terminal (or head). The length of the line segment is called a magnitude of the directed line segment. See Fig.(i). The same line segment with arrowhead in the opposite direction is also a directed line segment. See Fig.(ii).

Vector:

A quantity having both magnitude and direction is called a vector. Ex: displacement, velocity, acceleration, force, etc. Note that a directed line segment is a vector (Fig.(iii)), denoted as or simply as and read as 'vector ' or 'vector '.

The point P from where the vector starts is called initial point and the point Q where it ends is called its terminal point. The distance between P and Q is called the magnitude of , denoted as ||, or ||, or 'p'. The arrow indicates the direction of the vector.

Note :
(i) Vectors are represented by thick letters (ex: a, PQ) or letters with bar (ex: , ).
(ii) If a is a vector, then its length is denoted by |a|. It is always non-negative scalar.

A vector having the origin of the chosen co-ordinate system (either 2D or 3D) as initial point and a point P as terminal point is called the position vector of P in that co-ordinate system. It is denoted by (or ).

If P(x, y) is a point in a plane, then the magnitude of is given by:
|| = . Similarly, if P(x, y, z) is a point in a space, then the magnitude of is given by: || = .

Let OP &equals; r be a position vector of a point P.
If it makes angles of α, β, γ
with positive direction of X, Y, Z axes respectively,
then cos α, cos β, cos γ
are called the direction cosines (DCs) of the vector r.
The DCs are usually denoted by l, m, n respectively.
Draw perpendicular from P to X, Y and Z axes and let
A, B, C be the feet of the perpendiculars respectively.
Refer the adjacent figure.
From Δ OAP,
Similarly
The coordinates x, y, z of the point P may also be expressed as (lr, mr, nr).
The numbers lr, mr, nr which are proportional to the DCs l, m, n are called the direction rations (DRs) of the vector r.
The DRs are usually denoted by a, b, c respectively.

Zero vector: A vector whose initial and terminal points coincide, is called a zero vector or null vector. In other words, a vector that has zero magnitude is called a zero vector. It is denoted by or , , etc. The direction of the zero vector is indeterminate.

Unit vector: A vector whose magnitude is 1 unit (i.e., unity) is called a unit vector. The unit vector in the direction of a given vector is represented by and read as 'b cap' or 'b hat'.

The only purpose of unit vectors is to describe the direction. In Cartesian coordinate system, , and are the unit vectors that point along the x, y and z-axes respectively.

Co-initial vectors: The vectors having the same initial point are called co-initial vectors.

Equal vectors: Two vectors and are said to be equal, if they have the same magnitude (i.e., || = ||) and direction regardless of whether they have the same initial points or not. If and are equal vectors, then = .

Negative of a vector: A vector having same magnitude but in the opposite direction to a given vector is called the negative vector. For example, vector is negative of the vector and written as: = – . In order to find the negative of a vector, we merely reverse its arrowhead (direction).

Space vector:
If a1, a2, a3 are three real numbers, then the ordered triad
(a1, a2, a3) is called a space vector.
The real numbers a1, a2, a3
are called first, second and third components of the space vector (a1, a2, a3).Ex: The 3D co-ordinates of an object in space: x, y and z

Free vector:
A vector which is independent of its position is called a free vector. Ex: The direction of motion of a wave.

Localised vector:
If a is a vector and P is a point, then the ordered pair (P, a) is called localised vector at P. Ex: The position vector is a localised vector or a fixed vector.

Note:
If a is a vector and P is a point, then there exists a point Q such that a &equals; PQ.
The vector PQ is called localized vector of a at P.

Like and unlike vectors:
Two vectors are said to be like vectors if they have the same direction.
Ex: An express train overtaking a passenger train
Two vectors are said to be unlike vectors if they are in opposite directions.
Ex: Positive and negative co-ordinate axes

Addition of vectors by using head-to-tail method (or graphical method) takes large amount of time as it involves drawing of vectors on a scale and measurement of angles. More importantly, it does not allow algebraic operations that otherwise would give a simple solution. We can extend algebraic techniques to vectors, provided vectors are represented on a rectangular coordinate system. The magnitude and direction of a resultant vector can often be determined by use of trigonometric functions.

The magnitude of the resultant of two vectors and is: || = , where θ is the angle between the two vectors.

Two or more vectors are said to be parallel or collinear if they lie on a line or on parallel lines (irrespective of their magnitudes and directions).

Note:
(i) Let AB &equals; a, CD &equals; b be two vectors.
If A, B, C, D lie on a line or AB &par; CD, then a, b are parallel.
(ii) If a, b are parallel, then a, b have the same direction or opposite directions.

Relation between two collinear vectors:

If a and b are two parallel or collinear vectors,
then there exists a scalar k such that a &equals; kb
If a &equals; a1i + a2j + a3k and b &equals;
b1i + b2j + b3k
are two parallel or collinear vectors,
then

Vectors are said to be coplanar if they lie in a plane or in parallel planes.
Otherwise they are said to be non-coplanar. Note :
(i) If a, b are two vectors, then a, b are coplanar.
(ii) Collinear vectors are coplanar.
(iii) If a, b are two vectors, then a, b, O are coplanar.
(iv) Let a &equals; OA, b &equals; OB, c &equals; OC be three vectors.
If a, b are not collinear,
they generate a plane .
If c lies is the plane , then a, b, c are coplanar.
Otherwise they are non-coplanar.

Test of coplanarity of three points:

Let a and b be two given non-zero non-collinear vectors.
Then any vector r coplanar with a and b
can be uniquely expressed as r &equals; xa + yb for some scalars x and y.
The vectors a = a1i + a2j + a3k,
b = b1i + b2j + b3k,
c = c1i + c2j + c3k are coplanar iff

Let OA &equals; a, OB &equals; b be two non-zero vectors.
The value of ∠AOB which lies between 0° and 180° is called the angle between a and b. It is denoted by (a, b). Four cases are shown in adjacent figure.

Let a1, a2, a3, ....... an be n vectors and l1, l2, l3, l4, ..... ln be n scalars.
Then l1 a1 + l2 a2 + l3 a3 + .......... + ln an is called linear combination of a1, a2, a3, ......... an Note:
(i) 4a – 2b + c is a linear combination of a, b, c.
(ii) a, b are non-collinear vectors, then every vector in the plane determined by a pair of supports of a and b
can be expressed as linear combination of a and b in one and only one way.
(iii) Three vectors are coplanar, if and only if, one of them is a linear combination of the other two.

Let OA &equals; a, OB &equals; b, OC &equals; c, be three non-coplanar vectors.
Then the triad of vectors (a, b, c) is said to from a right-handed system
if the angle of rotation from OA to OB in anticlockwise direction
does not exceed 180° when observing from C.
Otherwise the triad of vectors is said to form a left-handed system.

Right-hand system of orthonormal vectors

A triad of three non-coplanar vectors i, j, k is said to be a right-hand system of orthonormal triad of vector if
(i) i, j, k form a right-handed system
(ii) i, j, k are unit vectors
(iii) (i, j) &equals; (j, k) &equals; (k, j) &equals; 90°

Note:

(i) The equation of the line passing through origin 'O' and parallel to the vector b is r &equals; t b, t &Element; R.
(ii) The Cartesian equation of a line passing through the point A (x1, y1, z1)
and parallel to the vector b &equals; l i + m j + n k is given by
(iii) The Cartesian equation of the line passing through the points A (x1, y1, z1) and B
(x2, y2, z2) is given by

(iv) The vector equation of the internal bisector of the angle between the vector b and c and
passing through the point a is r &equals; a t (b + c).
(v) The Cartesian equation of a plane passing through the point A (x1, y1, z1) and parallel to the vector b &equals; b1i + b2j + b3k and c &equals; c1i + c2j + c3k is given by